The Cake Icing Puzzle

Take a cake with icing on the top and no icing on the bottom. Pick an angle , cut a piece out of the cake in the usual way with center angle , turn the piece upside down, and replace it in the gap in the cake. Then do it again, with a piece that borders the initial piece. Keep doing this, moving in the same direction, thus forming cut lines at , , , , and so on.

1. Will it ever happen, after the first inverting operation, that the cake is in its initial position, with all of the icing on the top?2. Will it ever happen that all the icing is on the bottom?

Of course, if is a right angle, for example, then the answers are clearly YES and YES after eight and four cuts, respectively. But what if is not a rational multiple of , say radian?

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DETAILS

This puzzle first appeared in the 31st Moscow Mathematical Olympiad. The complete solution can be found in the book by P. Winkler, Mathematical Mind-Benders, Natick, MA: AK Peters, 2007. If is the smallest integer greater than or equal to , then the icing covers the top of the cake after iterations (unless , in which case it takes iterations). The icing will never all be on the bottom of the cake (unless , in which case this happens after iterations). Snapshot 3 shows the situation after 83 cuts for the case , , and ; one more operation restores all the icing.